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Sum-Of-Squares Bounds via Boolean Function Analysis

Adam Kurpisz, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We introduce a method for proving bounds on the SoS rank based on Boolean Function Analysis and Approximation Theory.  ...  Finally, we consider the question by Bienstock regarding the instance of the Set Cover problem. For this problem we prove the SoS rank lower bound of Ω( √ n).  ...  I C A L P 2 0 1 9 79:8 Sum-Of-Squares Bounds via Boolean Function Analysis Theorem 10.  ... 
doi:10.4230/lipics.icalp.2019.79 dblp:conf/icalp/Kurpisz19 fatcat:nwlhgdo6inav5cbj2fjacjuqti

Algorithm 959

José Antonio Álvarez-Cubero, Pedro J. Zufiria
2016 ACM Transactions on Mathematical Software  
Moreover, because of the size and complexity of modern ciphers, automatic analysis programs are very helpful in reducing the time required to study cryptographic properties of vector Boolean functions.  ...  In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate  ...  Search for Vector Boolean Functions with Excellent Profiles Boolean functions with very high nonlinearity pose some of the most challenging problems in the area of symmetric cryptography and combinatorics  ... 
doi:10.1145/2794077 fatcat:4tgey4sslzbxjfly4t3sra6hc4

Circuit and Decision Tree Complexity of Some Number Theoretic Problems

Anna Bernasconi, Carsten Damm, Igor Shparlinski
2001 Information and Computation  
We extend the area of applications of the Abstract Harmonic Analysis to lower bounds on the circuit and decision tree complexity of Boolean functions related to some number theoretic problems.  ...  In particular, we prove that deciding if a given integer is square-free and testing co-primality of two integers by unbounded fan-in circuits of bounded depth requires superpolynomial size.  ... where the sum is taken over all strings w 2 B n of Hamming weight jwj > #. u t Note that a decision tree of size M can be simulated by the an unbounded fan-in Boolean circuit of  ... 
doi:10.1006/inco.2000.3017 fatcat:bqqmlfrysffl3fb6675w2tk4jy

Sum-of-squares hierarchies for binary polynomial optimization [article]

Lucas Slot, Monique Laurent
2022 arXiv   pre-print
We consider the sum-of-squares hierarchy of approximations for the problem of minimizing a polynomial f over the boolean hypercube 𝔹^n={0,1}^n.  ...  This hierarchy provides for each integer r ∈ℕ a lower bound f_(r) on the minimum f_min of f, given by the largest scalar λ for which the polynomial f - λ is a sum-of-squares on 𝔹^n with degree at most  ...  Our objective in this paper is to investigate such an error analysis for this hierarchy applied to binary polynomial optimization as in (1) . 1.1. The sum-of-squares hierarchy on the boolean cube.  ... 
arXiv:2011.04027v3 fatcat:io3h2ofz2bdktkh4ct3ma7ufqe

Optimal quantum query bounds for almost all Boolean functions

Andris Ambainis, Arturs Backurs, Juris Smotrovs, Ronald De Wolf, Marc Herbstritt
2013 Symposium on Theoretical Aspects of Computer Science  
Our proof uses the fact that the acceptance probability of a T -query algorithm can be written as the sum of squares of degree-T polynomials.  ...  Optimal quantum query bounds for almost all Boolean functions Ambainis, A.; Bačkurs, A.; Smotrovs, J.; de Wolf, R.  ...  Acknowledgement We thank Loïck Magnin and Jérémie Roland for sending us a copy of [16] .  ... 
doi:10.4230/lipics.stacs.2013.446 dblp:conf/stacs/AmbainisBSW13 fatcat:ppztmbk3bnbplehbya7tf5ny7m

Widening ROBDDs with Prime Implicants [chapter]

Neil Kettle, Andy King, Tadeusz Strzemecki
2006 Lecture Notes in Computer Science  
The number of operations is, in turn, constrained by the number of times a Boolean function can be weakened before stability is achieved.  ...  Despite the ubiquity of ROBDDs in program analysis, and extensive literature on ROBDD minimisation, there is a dearth of work on approximating ROBDDs.  ...  Previous attempts at bounding the iterations have confined the analysis to a fixed sub-domain of Boolean formulae [13] .  ... 
doi:10.1007/11691372_7 fatcat:ijsr4atas5gwln4733t5admej4

Hilbert Function and Complexity Lower Bounds for Symmetric Boolean Functions

Anna Bernasconi, Lavinia Egidi
1999 Information and Computation  
This paper explores the application of certain algebraic geometry techniques involving Hilbert functions and Gro bner bases to the analysis of properties of Boolean functions.  ...  It gives some results and applications for symmetric functions. ]  ...  Similarly, in [1] , matrix analysis yields a lower bound for a coin weighing problem and, as a by-product, a generalization of a bound on weights for threshold gates.  ... 
doi:10.1006/inco.1999.2798 fatcat:ot5k7c37wbazxlqd2y2uj6mzuu

An O(nlog log n) Learning Algorithm for DNF under the Uniform Distribution

Y. Mansour
1995 Journal of computer and system sciences (Print)  
We show that a DNF with terms of size at most d can be approximated by a function with at most d O(d log1=") non zero Fourier coe cients such that the expected error squared, with respect to the uniform  ...  In this case our algorithm learns a DNF with a polynomial number of terms in time n O(loglog n) , and a DNF with terms of size at most O(log n= log log n) in polynomial time.  ...  Acknowledgements I would like to thank Eyal Kushilevitz for commenting on an early version of the paper.  ... 
doi:10.1006/jcss.1995.1043 fatcat:2xj62btz25bxzktokukrhhxusq

Polynomial-Time Probabilistic Reasoning with Partial Observations via Implicit Learning in Probability Logics

Brendan Juba
In this work we consider the use of bounded-degree fragments of the "sum-of-squares" logic as a probability logic.  ...  It is known that this logic is capable of deriving many bounds that are useful in probabilistic analysis. We show here that it furthermore captures key polynomial-time fragments of resolution.  ...  We thus observe that sum-of-squares is quite powerful for probabilistic analysis. We now turn to consider the Boolean reasoning ability of sum-of-squares.  ... 
doi:10.1609/aaai.v33i01.33017866 fatcat:qfdy6vwqrzdhdcyhs3y5szchvq

Computing the moments of k-bounded pseudo-Boolean functions over Hamming spheres of arbitrary radius in polynomial time

Andrew M. Sutton, L. Darrell Whitley, Adele E. Howe
2012 Theoretical Computer Science  
We show that given a k-bounded pseudo-Boolean function f , one can always compute the cth moment of f over regions of arbitrary radius in Hamming space in polynomial time using algebraic information from  ...  of the region is exponential in the problem size.  ...  The authors would like to thank Schloß Dagstuhl -Leibniz Center for Informatics where some of the ideas contained in this paper were developed during the Dagstuhl seminar on the Theory of Evolutionary  ... 
doi:10.1016/j.tcs.2011.02.006 fatcat:x2dzaivawzb2lee3k4wvjauhlu

Book announcements

1990 Discrete Applied Mathematics  
A characterization of a cone of pseudo-Boolean functions via supermodularity-type inequalities (Y. Crama, P.L. Hammer and R. Holzman).  ...  Contrasts, treatment comparisons and component sums of squares. Least squares estimators for linear models. Properties of least squares estimators. Overparameterisation and constraints.  ... 
doi:10.1016/0166-218x(90)90013-3 fatcat:5jdut3iylrffvcnyiogm2tfe24

Decision procedure based discovery of rare behaviors in Stochastic Differential Equation models of biological systems

Arup K. Ghosh, Faraz Hussain, Sumit K. Jha, Christopher J. Langmead, Susmit Jha
2012 2012 IEEE 2nd International Conference on Computational Advances in Bio and medical Sciences (ICCABS)  
Unfortunately, due to the limited availability of analytic methods for SDEs, stochastic simulations are the most common means for estimating (or bounding) the probability of rare behaviors.  ...  We also compute the probability of an observed behavior under the assumption of Gaussian noise.  ...  Lemma 4: Given a lower bound on the probability density P of a discretized SDE behavior with m samples every ∆ time apart, the sum of squares of increments of Brownian motion T ≡ m i=1 W ti − W ti−1 2  ... 
doi:10.1109/iccabs.2012.6182635 dblp:conf/iccabs/GhoshHJLJ12 fatcat:2evjoriuovf4rdth5qdkbqomdq

Quantum boolean functions [article]

Ashley Montanaro, Tobias J. Osborne
2010 arXiv   pre-print
In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f^2 = I.  ...  Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions.  ...  TJO was supported by the University of London central research fund.  ... 
arXiv:0810.2435v5 fatcat:suipqescfvhilnqvpy3n44dszy

Polynomial-time probabilistic reasoning with partial observations via implicit learning in probability logics [article]

Brendan Juba
2018 arXiv   pre-print
In this work we consider the use of bounded-degree fragments of the "sum-of-squares" logic as a probability logic.  ...  It is known that this logic is capable of deriving many bounds that are useful in probabilistic analysis. We show here that it furthermore captures useful polynomial-time fragments of resolution.  ...  For Boolean reasoning, sum-of-squares simulates a variety of fragments of systems for which polynomial-time algorithms were already known.  ... 
arXiv:1806.11204v1 fatcat:4yf5xyqnzbesri4i7bfwkwbzhi

A Brief Introduction to Fourier Analysis on the Boolean Cube

Ronald de Wolf
2008 Theory of Computing  
We give a brief introduction to the basic notions of Fourier analysis on the Boolean cube, illustrated and motivated by a number of applications to theoretical computer science.  ...  Several more extensive surveys on Fourier analysis of Boolean functions exist in the literature.  ...  Applying the KKL Inequality, for any δ ∈ [0, 1] we can bound the sum of squared biases by ∑ S∈( [n] k ) β 2 S = 2 2n |A| 2 ∑ S∈( [n] k ) f (S) 2 ≤ 2 2n δ k |A| 2 |A| 2 n 2/(1+δ ) ≤ 1 δ k 2 n |A| 2δ .  ... 
doi:10.4086/ dblp:journals/toc/Wolf08 fatcat:ijinjzubuzc4plzacz64xnn7na
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